Separation of transport in slow and fast time-scales using modulated heat pulse experiments (hysteresis in flux explained)

Old and recent experiments show that there is a direct response to the heating power of transport observed in modulated ECH experiments both in tokamaks and stellarators. This is most apparent for modulated experiments in the Large Helical Device (LHD) and in Wendelstein 7 advanced stellarator (W7-AS). In this paper we show that: (1) this power dependence can be reproduced by linear models and as such hysteresis (in flux) has no relationship to hysteresis as defined in the literature; (2) observations of 'hysteresis' (in flux) and a direct response to power can be perfectly reproduced by introducing an error in the estimated deposition profile as long as the errors redistribute the heat over a large radius; (3) non-local models depending directly on the heating power can also explain the experimentally observed Lissajous curves (hysteresis); (4) how non-locality and deposition errors can be recognized in experiments and how they affect estimates of transport coefficients; (5) from a linear perturbation transport experiment, it is not possible to discern deposition errors from non-local fast transport components (mathematically equivalent). However, when studied over different operating points non-linear-non-local transport models can be derived which should be distinguishable from errors in the deposition profile. To show all this, transport needs to be analyzed by separating the transport in a slow (diffusive) time-scale and a fast (heating/non-local) time-scale, which can only be done in the presence of perturbations. (paper)